Three-dimensional measurements of an inclined vortex ring interacting with a density stratification

Vortex rings are generated every time an impulsive jet occurs in an ambient homogeneous fluid, for instance in thermal or density plumes in environmental applications. Those vortical structures, often referenced as building blocks of turbulence, exist in a wide range of scales. The velocity field associated with these toroidal vortical structures makes them self-propagating. Such vortices are likely to carry mass and momentum along their path and are therefore good candidates for driving mixing to locations that are remote from where the ring is created. In numerous environmental applications, vortex rings interact with a density stratification due to variations of salinity and/or temperature. The path followed by a light homogeneous and axisymmetric vortex ring impinging a layer of fluid stably stratified in density with a given angle $\theta$ to the vertical, along with its dynamics and subsequent reorganization, are experimentally addressed in the present study. A three-dimensional time-resolved particle tracking velocimetry technique is set up to highlight the response of the stratified layer that is characterized by the generation of baroclinic vorticity and internal gravity waves. For the experimental parameters of the present study, the vorticity generation and reorganization, as well as the internal gravity waves in the stratified layer, are shown to remain axisymmetric for normal impacts. When the vortex ring propagation axis is not aligned with the density gradient, the expected symmetry breaking of the flow is observed. For inclined impact, the flow reorganizes in the form of a vertically flattened dipolar structure. In this case, internal gravity waves radiate away from two sources that match with the dipole cores.

Figure 1

Sketch (not to scale) of the experimental setup. (a) Complete view and (b) view from above.

Figure 2

Vertical density profile targeted (dashed black) and measured with the conductivity probe (red).

Figure 3

Sketches of the (a) 3D biplanar calibration target and (b) calibration setup. Figures are not to scale.

Figure 4

Example of disparity maps obtained on one of the four cameras (a) prior to the self-calibration step, with a mean magnitude of $1.88\text{pixels}$, and (b) after the mapping function correction, with a mean magnitude of $0.08\text{pixels}$.

Figure 5

Vortex-ring (a),(b) vertical and (c),(d) spanwise velocity in the vortex-ring vertical symmetry plane $y/d=0$ before a normal impact ($\theta =0^{\circ }$) with the linear stratification. (a),(c) 25 consecutive particle positions extracted from Lagrangian data on the whole volume depth ($y/d\in [-2.5,2.5]$) are plotted. (b),(d) Color maps of velocity are derived from Eulerian data, along with the quiver of the velocity $(u,w)$. Velocities and positions are nondimensionalized with the piston velocity $V_p$ and diameter $d$, respectively.

Figure 6

(a) Histogram of track length (sliding averaged on 10 frames for clarity) in semilogarithmic coordinates. (b) An example of the trajectory colored by the time. Joint PDF of $-\frac{dw}{dz}$ with $\frac{du}{dx}+\frac{dv}{dy}$, (c) before and (d) after vortex-ring interaction with the stratified layer.

Figure 7

(a) Sketch of the cylindrical sample volume used to compute the velocity profiles. Axis $s$ is the vortex-ring direction of propagation, the green box represents the lighted fluid volume, the red dashed lines at $s/d=0$ indicate the stratification upper boundary position, and the gray cylinder represents the volume used for the calculation in line with the vortex-ring launcher. At a given time, particles inside the gray cylinder of diameter $d$ are detected and their Lagrangian velocities are used to establish the mean velocity $V_s$ along $s$. Time evolution (color) of the velocity profiles $V_s$ along $s$ for (b) $\theta =0^{\circ }$ and (c) $\theta ={30}^{\circ }$. Lines are solid when $V_s>0$ (structure going downwards) and dashed when $V_s<0$. The top-right inset indicates the velocity extremum position as a function of time (square and circle are used for maximum and minimum, respectively). Note that a given time is associated with a specific color used both for the figure and its inset. $V_s$, $s$, and $t$ are normalized by $V_p$, $d$, and $d/V_p$, respectively.

Figure 8

Vortex ring impacting the stratification vertically at $t^{*}=[6,11,15,20]$ (columns from left to right). (a) The gray envelope indicates the isosurface of vorticity, $‖\omega ‖=0.5{\omega }_{max}$. (b) Color maps of the nondimensional $y$ component of vorticity ${\omega }_{y}d/V_p$ are shown in the vertical symmetry plane $y/d=0$, and gray lines are associated with the in-plane components of the velocity streamlines in the laboratory reference frame. The red dashed line marks the stratification upper boundary.

Figure 9

Schematic of the mechanism of baroclinic vorticity generation by a normal impact of a vortex ring on a density stratification.

Figure 10

Vortex ring impacting the stratification with an angle $\theta ={30}^{\circ }$ to the vertical at $t^{*}=[6,11,15,20]$ (columns from left to right). (a) Color maps show nondimensional horizontal vorticity in the vertical symmetry plane $y/d=0$. Solid thick black lines give the trajectory followed by the vortex ring. (b) Color maps represent nondimensional vertical vorticity in the horizontal plane $z/d=1.3$. Gray lines in rows (a) and (b) are associated with streamlines in the laboratory reference frame. The star in the top left figure points out a hyperbolic point at $(x/d,z/d)\approx (-1,2)$. The red dashed line in (a) marks the stratification upper boundary and the cyan solid line indicates the position of the horizontal plane $z/d=1.3$. The black dashed line in (b) marks the vertical symmetry plane $y/d=0$.

Figure 11

(a) Schematics of the mechanism of baroclinic vorticity generation by an inclined impact of a vortex ring on a density stratification. (b) Schematics of the subsequent vorticity organization after erosion and merging of the vortex ring with the baroclinic vorticity.

Figure 12

(a) Nondimensional vertical vorticity field ${\omega }_{z}d/V_p$ and (b) vertical velocity field $w/V_p$ in the horizontal plane $z/d=1$ and (c) horizontal velocity field $u/V_p$ in the vertical symmetry plane $y/d=0$ for an inclined vortex ring impact at $t^{*}=40$. For each figure, (a),(b) gray streamlines and (c) arrows correspond to the in-plane velocity field.

Figure 13

Characteristics of the vortex dipole at $t^{*}=45$ or, equivalently, at $Nt=28$. Horizontal velocity field $u/V_p$, (a) in the vertical symmetry plane $y/d=0$ and (d) in the horizontal plane $z/d=1.05$. The green dot located at $(x/d,y/d,z/d)=(2.85,0,1.05)$ marks the position of the horizontal velocity maximum. (c),(e),(b) Horizontal velocity profiles plotted along lines containing the green dot and directed by $x$ [blue line in (a) and (d)], $y$ [orange line in (d)], and $z$ [green line in (a)], respectively. Horizontal profiles are fitted with a Gaussian function centered on the maximum horizontal velocity position (black curves).

Figure 14

Time evolution of the vortex dipole characteristic dimensions (solid lines) and fit of the experimental data with a function $L_i/L_{i_0}=a_i\sqrt{\nu (t-b_i)}+c_{i}cos(d_{i}t+e_i)$ with $i=\{x,y,z\}$ (dashed lines).

Figure 15

Isosurfaces of (a) radial and (b) vertical velocity in the stratified zone subsequent to the vertical impact ($\theta =0^{\circ }$) of the vortex ring at $t^{*}=35$ (equivalently, $Nt=22$). Red isosurfaces correspond to (a) radial velocities from the line $x/d=y/d=0$ or (b) downward velocities, while blue isosurfaces correspond to velocities oriented the other way around. The red dashed line points to the stratification upper boundary $z/d=0$.

Figure 16

(a) Power spectrum of the velocity. After convolution of the FFT output with a bandpass filter, for each frequency band, the mean velocity angle $\alpha$ with respect to the vertical for $z/d\ge 0$ at $Nt=40$ is computed for (b) the normal impact and (e) the inclined one. Black lines show the analytical internal gravity wave dispersion relation for comparison. The vertical error bar extension is the frequency bandwidth $\mathrm{\Delta }\omega /N=0.1$, and the horizontal one is twice the angle standard deviation. (c)–(f) Velocity field filtered at the frequency $\omega /N=0.75$ in the vertical symmetry plane $y/d=0$ for the normal and inclined vortex-ring impacts, respectively. (d)–(g) Similar to (c) and (f) for $\omega /N=0.25$. The color field is the velocity magnitude normalized by the piston velocity. The red (blue) solid line is aligned with the internal wave beam angle expected for $\omega /N=0.25$ ($\omega /N=0.75$).

Figure 17

Spatiotemporal diagram of vertical velocity in the horizontal plane $z/d=2$ for the (a) normal and (c) inclined impact. (b) Time evolution of the vertical velocity for the normal impact at $(x/d,z/d)=(0,2)$ and three different spanwise locations. (d) Time evolution of the vertical velocity for the inclined impact at $(x/d,z/d)=(2,2)$ and three different spanwise locations.